💡Okay, so I have this problem here: the equation (2x^2 - 3x - 1 = 0) has one root as (m), and I need to find the value of the expression (3m(2m - 3) - 1). Hmm, let me think about how to approach this.First, I remember that if (m) is a root of the equation, then plugging (m) into the equation should satisfy it. So, substituting (x = m) into the equation gives:[2m^2 - 3m - 1 = 0]That's a good starting point. Now, I need to find the value of (3m(2m - 3) - 1). Let me write that expression down:[3m(2m - 3) - 1]I think I can simplify this expression step by step. Let me expand the multiplication first:First, multiply (3m) by (2m):[3m times 2m = 6m^2]Then, multiply (3m) by (-3):[3m times (-3) = -9m]So, putting it all together, the expression becomes:[6m^2 - 9m - 1]Now, I have the expression (6m^2 - 9m - 1). I need to find the value of this expression. But I already know from the quadratic equation that (2m^2 - 3m - 1 = 0). Maybe I can use this to simplify (6m^2 - 9m - 1).Let me see. If I multiply the entire quadratic equation by 3, what do I get?[3 times (2m^2 - 3m - 1) = 0]Calculating that:[6m^2 - 9m - 3 = 0]Wait, that's interesting. So, (6m^2 - 9m - 3 = 0). But my expression is (6m^2 - 9m - 1). Hmm, they look similar except for the constant term.Let me write down both:From the quadratic equation multiplied by 3:[6m^2 - 9m - 3 = 0]My expression:[6m^2 - 9m - 1]If I subtract these two equations, I can find the difference:[(6m^2 - 9m - 1) - (6m^2 - 9m - 3) = 0 - 0]Simplifying the left side:[6m^2 - 9m - 1 - 6m^2 + 9m + 3 = 0]The (6m^2) and (-6m^2) cancel out, and the (-9m) and (+9m) also cancel out. So, I'm left with:[-1 + 3 = 0]Which simplifies to:[2 = 0]Wait, that doesn't make sense. Did I make a mistake somewhere?Let me go back. I had the expression (6m^2 - 9m - 1), and from the quadratic equation, I had (6m^2 - 9m - 3 = 0). So, if I rearrange the quadratic equation:[6m^2 - 9m = 3]Then, substitute this into my expression:[6m^2 - 9m - 1 = 3 - 1 = 2]Oh, that's it! So, instead of subtracting the equations, I can directly substitute (6m^2 - 9m) with 3. Therefore, the expression simplifies to 2.Let me double-check to make sure I didn't make any mistakes. Starting from the quadratic equation:[2m^2 - 3m - 1 = 0]Multiply both sides by 3:[6m^2 - 9m - 3 = 0]So, (6m^2 - 9m = 3). Then, the expression (6m^2 - 9m - 1) becomes (3 - 1 = 2). Yep, that seems correct.I think I initially confused myself by trying to subtract the equations, which led to an incorrect conclusion. But by directly substituting, I get the right answer.So, the value of the expression (3m(2m - 3) - 1) is 2.

💡Okay, so I have this problem about a pizzeria blending cheeses. They have a special blend that costs 696.05 per kilogram. The blend is made up of mozzarella, romano, and provolone cheeses, which cost 504.35, 887.75, and 735.25 per kilogram respectively. The ratio of these cheeses in the blend is 2:1:2 for mozzarella, romano, and provolone.The employees have already shredded 20 kilograms of mozzarella and 10 kilograms of romano cheese. I need to find out how many kilograms of provolone cheese they need to shred now to maintain the correct proportion.First, I need to understand the ratio. The ratio is 2:1:2, which means for every 2 parts of mozzarella, there's 1 part of romano, and 2 parts of provolone. So, mozzarella and provolone are in equal proportions relative to romano.Given that they have already shredded 20 kg of mozzarella and 10 kg of romano, I can check if these quantities align with the given ratio. According to the ratio, mozzarella should be twice as much as romano. Here, 20 kg of mozzarella is indeed twice 10 kg of romano, so that part checks out.Now, since the ratio of mozzarella to provolone is 2:2, which simplifies to 1:1, that means the amount of mozzarella should equal the amount of provolone. Since they have already shredded 20 kg of mozzarella, they should also have 20 kg of provolone to maintain the ratio.Wait, but I'm supposed to calculate how much more provolone they need to shred now. They already have some amount of provolone, but the problem doesn't specify how much they've already shredded. It only mentions that they've shredded 20 kg of mozzarella and 10 kg of romano. So, if they need a total of 20 kg of provolone to match the 20 kg of mozzarella, and assuming they haven't shredded any provolone yet, they would need to shred 20 kg of provolone.But maybe I should double-check this by considering the total cost and ensuring that the blend's cost per kilogram matches 696.05. Let's see:If they have 20 kg of mozzarella at 504.35/kg, that's 20 * 504.35 = 10,087.10 kg of romano at 887.75/kg is 10 * 887.75 = 8,877.50.And if they have 20 kg of provolone at 735.25/kg, that's 20 * 735.25 = 14,705.Total cost would be 10,087 + 8,877.50 + 14,705 = 33,669.50.Total weight is 20 + 10 + 20 = 50 kg.Cost per kilogram would be 33,669.50 / 50 = 673.39, which is less than the required 696.05. Hmm, that's a problem.So, maybe my initial assumption that they need 20 kg of provolone is incorrect because the cost per kilogram doesn't match. I need to find the correct amount of provolone that, when combined with 20 kg of mozzarella and 10 kg of romano, results in a blend costing 696.05 per kilogram.Let me set up an equation for this. Let x be the amount of provolone needed.Total cost = (20 * 504.35) + (10 * 887.75) + (x * 735.25)Total weight = 20 + 10 + x = 30 + xCost per kilogram = Total cost / Total weight = 696.05So,(20 * 504.35 + 10 * 887.75 + x * 735.25) / (30 + x) = 696.05Calculating the total cost:20 * 504.35 = 10,08710 * 887.75 = 8,877.50So, total cost without provolone is 10,087 + 8,877.50 = 18,964.50Adding x * 735.25 gives total cost = 18,964.50 + 735.25xTotal weight = 30 + xSo,(18,964.50 + 735.25x) / (30 + x) = 696.05Multiply both sides by (30 + x):18,964.50 + 735.25x = 696.05 * (30 + x)Calculate 696.05 * 30 = 20,881.50696.05 * x = 696.05xSo,18,964.50 + 735.25x = 20,881.50 + 696.05xSubtract 18,964.50 from both sides:735.25x = 1,917 + 696.05xSubtract 696.05x from both sides:39.2x = 1,917Divide both sides by 39.2:x = 1,917 / 39.2 ≈ 48.85 kgWait, that can't be right because they already have 20 kg of mozzarella and 10 kg of romano, which is 30 kg total. Adding 48.85 kg of provolone would make the total weight 78.85 kg, which seems excessive. Also, the ratio would be way off because mozzarella is only 20 kg and provolone is 48.85 kg, which doesn't match the 2:1:2 ratio.I think I made a mistake in setting up the equation. The ratio should be maintained, so the amounts of each cheese should be in the ratio 2:1:2. Since they have 20 kg of mozzarella and 10 kg of romano, which is a 2:1 ratio, the amount of provolone should be 20 kg to maintain the 2:1:2 ratio. But when I calculated the cost per kilogram with 20 kg of provolone, it didn't match 696.05. So, there's a conflict here.Maybe the ratio isn't by weight but by volume or something else, but the problem states it's a proportion, so I think it's by weight. Alternatively, perhaps the cost per kilogram of the blend is an average based on the ratio, so I need to calculate the weighted average cost.Let me try that approach. The blend has a ratio of 2:1:2, which sums to 5 parts. So, mozzarella is 2/5, romano is 1/5, and provolone is 2/5 of the blend by weight.The cost per kilogram of the blend should be:(2/5)*504.35 + (1/5)*887.75 + (2/5)*735.25Calculate each term:(2/5)*504.35 = 201.74(1/5)*887.75 = 177.55(2/5)*735.25 = 294.10Add them up: 201.74 + 177.55 + 294.10 = 673.39But the blend is supposed to cost 696.05 per kilogram, which is higher than 673.39. This suggests that the ratio might not be by weight but perhaps by volume or another measure, or there's an error in the problem statement.Alternatively, maybe the ratio is by volume, and the densities of the cheeses are different, but the problem doesn't provide that information. So, I'm stuck.Given that, I'll proceed with the initial assumption that the ratio is by weight, and they need 20 kg of provolone to match the 20 kg of mozzarella and 10 kg of romano. Even though the cost per kilogram doesn't match, perhaps there's a miscalculation or the problem expects us to ignore the cost discrepancy and focus on the ratio.So, the answer is 20 kg of provolone.

💡Okay, so I have this functional equation to solve: ( f(f(x) + y) = 2x + f(f(y) - x) ) for all real numbers ( x ) and ( y ). Hmm, functional equations can be tricky, but let me try to break it down step by step.First, I remember that for functional equations, a common strategy is to substitute specific values for ( x ) or ( y ) to simplify the equation and maybe find some properties of the function ( f ). Let me see what happens if I choose some specific values.Let me try setting ( y = -f(x) ). That might simplify the left side. So, substituting ( y = -f(x) ) into the equation gives:( f(f(x) + (-f(x))) = 2x + f(f(-f(x)) - x) )Simplifying the left side, ( f(0) = 2x + f(f(-f(x)) - x) ). Okay, so this tells me that ( f(0) ) is equal to ( 2x + f(f(-f(x)) - x) ). Hmm, interesting. Since this equation holds for all real numbers ( x ), the right side must be a constant because the left side is just ( f(0) ), which is a constant. That suggests that ( 2x + f(f(-f(x)) - x) ) is constant for all ( x ). Wait, if ( 2x + f(f(-f(x)) - x) ) is constant, then the term ( f(f(-f(x)) - x) ) must be equal to ( -2x + c ), where ( c ) is a constant. That is, ( f(f(-f(x)) - x) = -2x + c ). But I don't know what ( c ) is yet. Maybe I can find another substitution or find more properties of ( f ). Let me think about whether ( f ) is injective or surjective. If ( f ) is injective, that would mean that if ( f(a) = f(b) ), then ( a = b ). If ( f ) is surjective, then for every real number ( y ), there exists an ( x ) such that ( f(x) = y ). Looking back at the equation ( f(0) = 2x + f(f(-f(x)) - x) ), since this holds for all ( x ), the right side must take all real values as ( x ) varies, but the left side is a constant. Wait, that seems contradictory. If ( 2x + f(f(-f(x)) - x) ) is equal to a constant for all ( x ), then ( f(f(-f(x)) - x) ) must be equal to ( -2x + c ), as I thought before. But if ( f(f(-f(x)) - x) = -2x + c ), then ( f ) must be surjective because for any real number ( y ), we can set ( y = -2x + c ), which implies that ( x = (c - y)/2 ), so ( f ) can take any real value. Therefore, ( f ) is surjective.Since ( f ) is surjective, that means there exists some real number ( a ) such that ( f(a) = 0 ). Let me substitute ( x = a ) into the original equation to see what happens.Substituting ( x = a ), we get:( f(f(a) + y) = 2a + f(f(y) - a) )But ( f(a) = 0 ), so this simplifies to:( f(0 + y) = 2a + f(f(y) - a) )Which is:( f(y) = 2a + f(f(y) - a) )Hmm, interesting. Let me rearrange this equation:( f(y) - 2a = f(f(y) - a) )This suggests that if I let ( z = f(y) - a ), then ( f(z) = f(y) - 2a ). But ( z = f(y) - a ), so ( f(z) = z - a ). Wait, that might not be directly helpful. Let me think again.Since ( f ) is surjective, for any real number ( w ), there exists a ( y ) such that ( f(y) = w ). So, substituting ( w = f(y) ), the equation becomes:( w = 2a + f(w - a) )Which can be rearranged as:( f(w - a) = w - 2a )Let me make a substitution here. Let ( u = w - a ), which implies ( w = u + a ). Substituting back, we get:( f(u) = (u + a) - 2a = u - a )So, ( f(u) = u - a ) for all real numbers ( u ). That suggests that ( f ) is a linear function of the form ( f(x) = x - a ).Let me verify if this function satisfies the original equation. Let's substitute ( f(x) = x - a ) into the equation:Left side: ( f(f(x) + y) = f((x - a) + y) = f(x + y - a) = (x + y - a) - a = x + y - 2a )Right side: ( 2x + f(f(y) - x) = 2x + f((y - a) - x) = 2x + f(y - a - x) = 2x + (y - a - x - a) = 2x + y - a - x - a = x + y - 2a )Both sides are equal, so the function ( f(x) = x - a ) indeed satisfies the original equation. Therefore, the solution is ( f(x) = x - a ) for some real constant ( a ).I think that's it! I managed to find the function by making strategic substitutions and using the properties of surjectivity. It's always good to check the solution in the original equation to make sure it works, and in this case, it does.

💡Okay, so I have this problem here: Point P is moving along the parabola C, which is given by the equation y = 2x². F is the focus of this parabola, and I need to find the minimum value of the distance |PF|. The options are A: 1, B: 1/2, C: 1/4, D: 1/8. Hmm, let me think about how to approach this.First, I remember that for a parabola, the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. But I'm not sure if that's directly helpful here. Maybe I should recall the standard form of a parabola and its properties.The given equation is y = 2x². I think the standard form of a parabola that opens upwards is y = (1/(4p))x², where p is the distance from the vertex to the focus. Let me check that. If I compare y = 2x² with y = (1/(4p))x², then 2 must be equal to 1/(4p). So, solving for p, I get 4p = 1/2, which means p = 1/8. So the focus F is located at (0, p), which would be (0, 1/8). Okay, that seems right.Now, the vertex of the parabola is at (0, 0). Since the parabola opens upwards, the vertex is the closest point to the focus. So, the distance from the vertex to the focus is p, which is 1/8. Therefore, the minimum distance |PF| should be 1/8, which is option D.Wait, but let me make sure I'm not making a mistake here. Maybe I should derive the distance formula and confirm. Let's take a general point P on the parabola, which can be represented as (x, 2x²). The focus F is at (0, 1/8). So, the distance |PF| can be calculated using the distance formula:|PF| = sqrt[(x - 0)² + (2x² - 1/8)²]To find the minimum value, I can minimize this distance. Since the square root function is increasing, I can instead minimize the square of the distance to make it simpler:D = x² + (2x² - 1/8)²Now, let's expand this:D = x² + (4x⁴ - (1/2)x² + 1/64)So, D = 4x⁴ + x² - (1/2)x² + 1/64Simplify the terms:D = 4x⁴ + (1 - 1/2)x² + 1/64D = 4x⁴ + (1/2)x² + 1/64Now, to find the minimum, I can take the derivative of D with respect to x and set it equal to zero.dD/dx = 16x³ + xSet dD/dx = 0:16x³ + x = 0x(16x² + 1) = 0So, x = 0 or 16x² + 1 = 0Since 16x² + 1 = 0 has no real solutions, the only critical point is at x = 0.Now, let's check the second derivative to confirm if this is a minimum.d²D/dx² = 48x² + 1At x = 0, d²D/dx² = 1, which is positive, so x = 0 is indeed a point of local minimum.Therefore, the minimum distance occurs at x = 0, which is the vertex of the parabola. Plugging x = 0 into the distance formula:|PF| = sqrt[(0)² + (0 - 1/8)²] = sqrt[0 + ( -1/8)²] = sqrt[1/64] = 1/8So, the minimum value of |PF| is 1/8, which is option D. That matches my initial thought.Just to be thorough, let me consider another approach. I remember that in a parabola, the distance from any point to the focus is equal to the distance from that point to the directrix. The directrix of the parabola y = 2x² is y = -1/8, right? Because the directrix is located at a distance p below the vertex, so y = -p = -1/8.So, for any point P(x, y) on the parabola, the distance to the focus F(0, 1/8) is equal to the distance to the directrix y = -1/8. The distance from P to the directrix is |y - (-1/8)| = |y + 1/8|. Since y = 2x², this becomes |2x² + 1/8|.But since 2x² is always non-negative, 2x² + 1/8 is always positive, so the distance is just 2x² + 1/8.Therefore, the distance |PF| is equal to 2x² + 1/8. To find the minimum value, we can minimize this expression.Since 2x² is always non-negative, the minimum occurs when x = 0, giving |PF| = 0 + 1/8 = 1/8.This confirms the previous result. So, the minimum value is indeed 1/8, which is option D.I think I've covered this problem from multiple angles: using calculus to find the minimum, recalling the definition of a parabola involving the focus and directrix, and considering the geometric properties of the parabola. All approaches lead me to the same conclusion that the minimum distance is 1/8.Just to make sure I didn't make any calculation errors, let me double-check the derivative part.Starting with D = 4x⁴ + (1/2)x² + 1/64First derivative: dD/dx = 16x³ + xSetting to zero: 16x³ + x = 0 => x(16x² + 1) = 0Solutions: x = 0 or 16x² + 1 = 0. The latter has no real solutions, so only x = 0 is critical.Second derivative: d²D/dx² = 48x² + 1, which is positive for all x, so x = 0 is a minimum.Plugging x = 0 into D: D = 0 + 0 + 1/64 = 1/64, so |PF| = sqrt(1/64) = 1/8.Yes, that's correct.Alternatively, using the distance formula directly:|PF| = sqrt(x² + (2x² - 1/8)^2)At x = 0, this becomes sqrt(0 + (0 - 1/8)^2) = sqrt(1/64) = 1/8.Everything checks out. So, I'm confident that the answer is D: 1/8.**Final Answer**boxed{D}

💡Okay, so I have this problem about rational points on the plane. A rational point is one where both the x and y coordinates are rational numbers. The task is to divide all these rational points into three mutually disjoint sets. These sets need to satisfy two conditions:1. Any circle centered at any rational point must contain three points, each from a different set.2. No three points on any straight line can belong to these three different sets.Alright, let me try to unpack this.First, I need to understand what it means to divide the rational points into three sets. They have to be mutually disjoint, so no point can be in more than one set. That makes sense. Now, the first condition is about circles. If I pick any rational point as the center of a circle, then no matter how big or small the circle is, it must contain at least one point from each of the three sets. So, in any circle around a rational center, there must be points from all three sets.The second condition is about straight lines. It says that on any straight line, you can't have three points, each from a different set. So, if I have a line, and I pick three points on that line, they can't all be from different sets. That seems like it's preventing the sets from being too "mixed" along lines.Hmm, okay. So, I need to find a way to categorize all rational points into three groups where circles around any rational point will always have at least one from each group, but lines won't have three from different groups.Let me think about how to approach this. Maybe I can use some properties of rational numbers or coordinates. Since both coordinates are rational, they can be expressed as fractions. Maybe I can use the denominators or numerators in some way.Wait, another thought: maybe I can use modular arithmetic. Since we're dealing with integers when we express the coordinates as fractions, perhaps I can categorize points based on the parity (evenness or oddness) of their numerators or denominators.Let me try that. Suppose I have a rational point (u/w, v/w), where u, v, w are integers, and w > 0, and the fraction is in lowest terms, so gcd(u, v, w) = 1.If I can categorize these points based on whether u, v, or w are even or odd, maybe that can help. Let's see.Define three sets:- Set A: Points where u is odd.- Set B: Points where u is even and v is odd.- Set C: Points where both u and v are even.Wait, but in this case, if u and v are both even, then w must be odd because gcd(u, v, w) = 1. So, in Set C, w is necessarily odd.But does this satisfy the conditions? Let me check.First, condition (2): No three points on any straight line can belong to these three different sets.Suppose I have a straight line with three points, each from A, B, and C. Let's see if that's possible.A line can be expressed as ax + by + c = 0, where a, b, c are integers with gcd(a, b, c) = 1. If a point (u/w, v/w) is on this line, then au + bv + cw = 0.If a point is in Set A, then u is odd. If it's in Set B, u is even and v is odd. If it's in Set C, both u and v are even.Suppose we have three points on the line, one from each set. Let's denote them as (u1/w1, v1/w1) from A, (u2/w2, v2/w2) from B, and (u3/w3, v3/w3) from C.So, for each point, we have:1. For Set A: u1 is odd.2. For Set B: u2 is even, v2 is odd.3. For Set C: u3 is even, v3 is even.But since all these points lie on the same line, the equation au + bv + cw = 0 must hold for each.Let me consider the parity of each term. For Set A: u1 is odd, so au1 is a times odd. If a is even, then au1 is even; if a is odd, au1 is odd. Similarly, bv1: if b is even, bv1 is even; if b is odd, bv1 is odd. cw1: since w1 is part of the point in Set A, which is (u1/w1, v1/w1). But in Set A, u1 is odd, and since gcd(u1, v1, w1) = 1, w1 must be odd (because if w1 were even, then u1 and w1 would both be odd, but v1 could be even or odd). Wait, actually, w1 could be even or odd, depending on v1.Wait, maybe this is getting too complicated. Let me think differently.If I have three points on a line, each from different sets, then their coordinates have different parities. Maybe this leads to a contradiction in the line equation.Alternatively, perhaps I can use the fact that if a line contains points from all three sets, then the line equation would have to satisfy certain parity conditions, which might not be possible.Alternatively, maybe I can think about scaling the coordinates. Since all coordinates are rational, I can scale them up to integers. So, for a point (u/w, v/w), I can think of it as (u, v) scaled by 1/w.But I'm not sure if that helps directly.Wait, another idea: maybe use coloring based on residues modulo 3 or something. But the problem specifies three sets, so maybe modulo 3 could work.But I'm not sure. Let me go back to the initial idea of using parity.If I define the sets based on the parity of u and v, as I did before, then maybe I can show that no line can have three points from all three sets.Suppose, for contradiction, that there is a line with points from A, B, and C. Then, considering the line equation au + bv + cw = 0, and the parities of u, v, w for each point, maybe I can find a contradiction.For example, take a point from Set A: u is odd, v can be anything, but since it's in lowest terms, w must be such that gcd(u, v, w) = 1. Similarly for other sets.But I'm not sure how to proceed with this. Maybe I need to look for a different approach.Wait, another thought: maybe use the fact that rational points are dense on the plane. So, any circle around a rational point will contain infinitely many rational points. Therefore, if I can ensure that among any three points in the circle, they are from different sets, that might work.But how to ensure that? Maybe by coloring the points in such a way that any circle contains points of all three colors.Alternatively, maybe use a checkerboard pattern, but extended to three colors.Wait, actually, I recall something about coloring the plane with three colors such that any circle contains all three colors, and any line doesn't have all three colors. Maybe that's related.But I'm not sure about the exact method. Maybe I can use a function based on coordinates to assign colors.Wait, another idea: use the denominators of the coordinates. Since any rational number can be written as a fraction with denominator w, maybe categorize based on w modulo 3.But I'm not sure if that would satisfy the conditions.Alternatively, maybe use the numerators modulo 3. So, for a point (u/w, v/w), if u ≡ 0 mod 3, put it in Set A; if u ≡ 1 mod 3, Set B; if u ≡ 2 mod 3, Set C.But then, would this satisfy the conditions? Let me see.For condition (1): Any circle around a rational point must contain points from all three sets. Since the rationals are dense, any circle will contain points with u ≡ 0, 1, 2 mod 3. So, that might work.For condition (2): No three points on a line can be from all three sets. Hmm, is that necessarily true? If I have a line, can I have three points with u ≡ 0, 1, 2 mod 3?Wait, maybe not, because the line equation would impose some constraints on u and v, which might restrict the residues.But I'm not sure. Maybe this approach doesn't guarantee condition (2).Wait, perhaps I need a different coloring. Maybe based on both u and v.Alternatively, think about the coordinates as vectors and use some linear algebra approach.Wait, another idea: use the concept of equivalence classes. Maybe partition the rational points based on some equivalence relation that ensures the conditions.But I'm not sure.Wait, going back to the initial idea of using parity. Let me try to formalize it.Define Set A: points where u is odd.Set B: points where u is even and v is odd.Set C: points where both u and v are even.Now, check condition (2): No three points on a line can be from all three sets.Suppose, for contradiction, that there is a line with points from A, B, and C.Let’s denote the line as ax + by + c = 0, with a, b, c integers, gcd(a, b, c) = 1.Take a point from Set A: (u1/w1, v1/w1), where u1 is odd.From Set B: (u2/w2, v2/w2), where u2 is even, v2 is odd.From Set C: (u3/w3, v3/w3), where u3 is even, v3 is even.Since all these points lie on the line, we have:a(u1/w1) + b(v1/w1) + c = 0a(u2/w2) + b(v2/w2) + c = 0a(u3/w3) + b(v3/w3) + c = 0Multiply through by w1, w2, w3 respectively:a u1 + b v1 + c w1 = 0a u2 + b v2 + c w2 = 0a u3 + b v3 + c w3 = 0Now, consider the parities.For the first equation: a u1 is a times odd. If a is even, then a u1 is even; if a is odd, a u1 is odd. Similarly, b v1: if b is even, even; if b is odd, odd. c w1: since w1 is part of the point in Set A, which is (u1/w1, v1/w1). Since u1 is odd and gcd(u1, v1, w1) = 1, w1 must be odd (because if w1 were even, then u1 and w1 would both be odd, but v1 could be even or odd). Wait, actually, w1 could be even or odd, depending on v1.Wait, maybe this is getting too complicated. Let me think differently.If I have three points on the line, each from different sets, then their coordinates have different parities. Maybe this leads to a contradiction in the line equation.Alternatively, perhaps I can use the fact that if a line contains points from all three sets, then the line equation would have to satisfy certain parity conditions, which might not be possible.Wait, another approach: consider the line equation modulo 2.If I have points from Set A, B, and C, then their coordinates modulo 2 would be:- Set A: (1, 0) or (1, 1)- Set B: (0, 1)- Set C: (0, 0)But actually, Set A includes points where u is odd, so modulo 2, u ≡ 1. v can be anything, so v ≡ 0 or 1.Set B: u ≡ 0, v ≡ 1.Set C: u ≡ 0, v ≡ 0.So, modulo 2, the points are:- Set A: (1, 0) or (1, 1)- Set B: (0, 1)- Set C: (0, 0)Now, if a line contains points from all three sets, then modulo 2, the line would have to pass through (1, 0), (1, 1), (0, 1), and (0, 0). But a line in the affine plane over GF(2) can have at most two points. So, it's impossible for a line to pass through all four points, which are the four possible points in GF(2)^2.Therefore, any line can contain at most two points from different sets. Hence, condition (2) is satisfied.Wait, that seems promising. So, by considering the line equation modulo 2, we see that a line can't contain points from all three sets because it would require the line to pass through all four points in GF(2)^2, which is impossible.Okay, so condition (2) is satisfied.Now, condition (1): Any circle centered at a rational point must contain points from all three sets.Since the rational points are dense in the plane, any circle around a rational center will contain infinitely many rational points. We need to ensure that among these, there are points from all three sets.Given that the sets are defined based on the parity of u and v, and since the rationals are dense, we can always find points with u odd, u even and v odd, and u even and v even within any circle.Therefore, condition (1) is satisfied.So, in conclusion, defining the sets based on the parity of the numerators u and v in the reduced form of the rational coordinates satisfies both conditions.I think this approach works. It uses the properties of rational points and modular arithmetic to ensure the required conditions are met.

💡Okay, so I have this problem where there are four identical point charges at the corners of a square, and the total energy stored in this configuration is 20 Joules. Then, one of these charges is moved to the center of the square, and I need to find out how much more energy is stored in the new configuration.First, I need to understand how the energy is stored in the initial configuration. Since all four charges are identical and placed at the corners of a square, the system has a certain symmetry. I know that the electric potential energy between two point charges is given by Coulomb's law:[ U = frac{k q_1 q_2}{r} ]where ( k ) is Coulomb's constant, ( q_1 ) and ( q_2 ) are the charges, and ( r ) is the distance between them.In the initial configuration, each charge interacts with the other three charges. However, since the charges are at the corners of a square, the distances between them are not all the same. Specifically, each charge has two neighbors at a distance equal to the side length of the square, let's call it ( s ), and one diagonal neighbor at a distance of ( ssqrt{2} ).But wait, the problem states that the total energy stored is 20 Joules. I need to figure out how this total energy is distributed among all the pairs of charges.There are four charges, so the number of unique pairs is ( binom{4}{2} = 6 ). Each pair contributes some amount of energy based on their distance apart.However, in a square, the distances between charges are not all unique. Specifically, there are two different distances: the side length ( s ) and the diagonal ( ssqrt{2} ).Let me count how many pairs are at each distance:- For each charge, there are two adjacent charges at distance ( s ), and one charge at the diagonal distance ( ssqrt{2} ). Since there are four charges, this might seem like there are 4 * 3 = 12 interactions, but since each interaction is counted twice (once for each charge), the actual number of unique pairs is 6.So, out of these 6 pairs, how many are at distance ( s ) and how many are at distance ( ssqrt{2} )?Looking at the square, each side has a length ( s ), and there are four sides, but each side is shared by two charges, so the number of unique pairs at distance ( s ) is 4.Similarly, the diagonals of the square are ( ssqrt{2} ), and there are two diagonals in a square, so the number of unique pairs at distance ( ssqrt{2} ) is 2.Therefore, in the initial configuration, there are 4 pairs at distance ( s ) and 2 pairs at distance ( ssqrt{2} ).Given that the total energy is 20 Joules, I can write:[ 4U_s + 2U_d = 20 ]where ( U_s ) is the energy between two charges at distance ( s ), and ( U_d ) is the energy between two charges at distance ( ssqrt{2} ).Since all charges are identical, ( U_s = frac{k q^2}{s} ) and ( U_d = frac{k q^2}{ssqrt{2}} ).So, substituting these into the equation:[ 4left(frac{k q^2}{s}right) + 2left(frac{k q^2}{ssqrt{2}}right) = 20 ]Let me factor out ( frac{k q^2}{s} ):[ frac{k q^2}{s} left(4 + frac{2}{sqrt{2}}right) = 20 ]Simplify ( frac{2}{sqrt{2}} ):[ frac{2}{sqrt{2}} = sqrt{2} ]So the equation becomes:[ frac{k q^2}{s} (4 + sqrt{2}) = 20 ]Let me denote ( frac{k q^2}{s} = U_s ), so:[ U_s (4 + sqrt{2}) = 20 ]Therefore:[ U_s = frac{20}{4 + sqrt{2}} ]To rationalize the denominator:[ U_s = frac{20}{4 + sqrt{2}} times frac{4 - sqrt{2}}{4 - sqrt{2}} = frac{20(4 - sqrt{2})}{(4)^2 - (sqrt{2})^2} = frac{20(4 - sqrt{2})}{16 - 2} = frac{20(4 - sqrt{2})}{14} ]Simplify:[ U_s = frac{20}{14}(4 - sqrt{2}) = frac{10}{7}(4 - sqrt{2}) ]So, ( U_s = frac{40 - 10sqrt{2}}{7} ) Joules.Similarly, ( U_d = frac{k q^2}{ssqrt{2}} = frac{U_s}{sqrt{2}} = frac{40 - 10sqrt{2}}{7sqrt{2}} )But maybe I don't need to find ( U_d ) explicitly right now.Now, moving on to the new configuration where one charge is moved to the center of the square.In this new configuration, there are still four charges, but one is at the center, and the other three are at the vertices.I need to calculate the total energy stored in this new configuration.First, let's figure out the distances between the charges.The center of the square is at a distance of ( frac{s}{sqrt{2}} ) from each vertex.So, the charge at the center is at distance ( frac{s}{sqrt{2}} ) from each of the three vertex charges.Additionally, the three vertex charges are still at distances ( s ) from their adjacent vertex charges, and ( ssqrt{2} ) from the opposite vertex charge.Wait, but in the new configuration, one charge is missing from a vertex, so the number of interactions changes.Wait, no, actually, all four charges are still present, but one is at the center, and the other three are at the vertices.So, the interactions are:1. Between the center charge and each of the three vertex charges: 3 interactions, each at distance ( frac{s}{sqrt{2}} ).2. Between the three vertex charges: Each pair of vertex charges can be adjacent or diagonal.But since one charge is moved to the center, the remaining three vertex charges form a sort of "L" shape, right? So, two of them are adjacent, and one is diagonal from the other two.Wait, no, actually, in a square, if one charge is moved to the center, the remaining three are still at the vertices, but one vertex is empty.So, the three vertex charges are still at the corners of the square, but one corner is empty.Therefore, the distances between the three vertex charges are:- Two adjacent pairs at distance ( s ).- One diagonal pair at distance ( ssqrt{2} ).So, in total, the interactions between the three vertex charges are:- Two pairs at distance ( s ).- One pair at distance ( ssqrt{2} ).Therefore, the total energy in the new configuration is the sum of:1. The energy between the center charge and each of the three vertex charges: 3 * ( U_{center} ), where ( U_{center} = frac{k q^2}{frac{s}{sqrt{2}}} = frac{k q^2 sqrt{2}}{s} = sqrt{2} U_s ).2. The energy between the three vertex charges: 2 * ( U_s ) + 1 * ( U_d ).So, total energy ( U_{total} ) is:[ U_{total} = 3 times sqrt{2} U_s + 2 U_s + U_d ]But from earlier, we have:[ 4 U_s + 2 U_d = 20 ]So, ( 2 U_d = 20 - 4 U_s ), which implies ( U_d = 10 - 2 U_s ).Substituting back into the expression for ( U_{total} ):[ U_{total} = 3 sqrt{2} U_s + 2 U_s + (10 - 2 U_s) ]Simplify:[ U_{total} = 3 sqrt{2} U_s + 2 U_s + 10 - 2 U_s ]The ( 2 U_s ) and ( -2 U_s ) cancel out:[ U_{total} = 3 sqrt{2} U_s + 10 ]Now, substitute ( U_s = frac{40 - 10sqrt{2}}{7} ):[ U_{total} = 3 sqrt{2} times frac{40 - 10sqrt{2}}{7} + 10 ]Let me compute this step by step.First, compute ( 3 sqrt{2} times (40 - 10sqrt{2}) ):[ 3 sqrt{2} times 40 = 120 sqrt{2} ][ 3 sqrt{2} times (-10sqrt{2}) = -30 times 2 = -60 ]So, the numerator becomes:[ 120 sqrt{2} - 60 ]Therefore:[ U_{total} = frac{120 sqrt{2} - 60}{7} + 10 ]Convert 10 to a fraction with denominator 7:[ 10 = frac{70}{7} ]So:[ U_{total} = frac{120 sqrt{2} - 60 + 70}{7} = frac{120 sqrt{2} + 10}{7} ]Simplify:[ U_{total} = frac{10(12 sqrt{2} + 1)}{7} ]But let's see if we can simplify this further or express it differently.Alternatively, let's compute the numerical value to understand the increase in energy.First, compute ( U_s ):[ U_s = frac{40 - 10sqrt{2}}{7} approx frac{40 - 14.142}{7} = frac{25.858}{7} approx 3.694 text{ Joules} ]Then, ( U_{total} = 3 sqrt{2} U_s + 10 approx 3 times 1.414 times 3.694 + 10 approx 3 times 1.414 times 3.694 approx 15.67 + 10 = 25.67 text{ Joules} )But wait, the initial total energy was 20 Joules, so the increase is approximately 5.67 Joules.But let's see if we can express this exactly.From earlier:[ U_{total} = frac{120 sqrt{2} + 10}{7} ]So, the increase in energy is:[ Delta U = U_{total} - 20 = frac{120 sqrt{2} + 10}{7} - 20 = frac{120 sqrt{2} + 10 - 140}{7} = frac{120 sqrt{2} - 130}{7} ]But this seems a bit messy. Maybe there's a better way to approach this.Alternatively, let's consider that when we move one charge to the center, we are adding new interactions and removing some old ones.In the initial configuration, the total energy is 20 Joules, which includes all 6 pairs.In the new configuration, we have:- 3 new pairs between the center charge and the three vertex charges.- The remaining interactions are between the three vertex charges, which are 3 pairs: two at distance ( s ) and one at distance ( ssqrt{2} ).Wait, but in the initial configuration, there were 4 pairs at distance ( s ) and 2 pairs at distance ( ssqrt{2} ).In the new configuration, we have:- 3 pairs at distance ( frac{s}{sqrt{2}} ).- 2 pairs at distance ( s ).- 1 pair at distance ( ssqrt{2} ).So, the total energy in the new configuration is:[ 3 times frac{k q^2}{frac{s}{sqrt{2}}} + 2 times frac{k q^2}{s} + 1 times frac{k q^2}{ssqrt{2}} ]Simplify:[ 3 sqrt{2} times frac{k q^2}{s} + 2 times frac{k q^2}{s} + frac{1}{sqrt{2}} times frac{k q^2}{s} ]Factor out ( frac{k q^2}{s} ):[ left(3 sqrt{2} + 2 + frac{1}{sqrt{2}}right) times frac{k q^2}{s} ]Combine the terms:[ left(3 sqrt{2} + 2 + frac{sqrt{2}}{2}right) times frac{k q^2}{s} ]Convert ( 3 sqrt{2} ) to ( frac{6 sqrt{2}}{2} ) and ( frac{sqrt{2}}{2} ) remains as is:[ left(frac{6 sqrt{2}}{2} + 2 + frac{sqrt{2}}{2}right) = left(frac{7 sqrt{2}}{2} + 2right) ]So, the total energy is:[ left(frac{7 sqrt{2}}{2} + 2right) times frac{k q^2}{s} ]But from earlier, we have:[ 4 times frac{k q^2}{s} + 2 times frac{k q^2}{s sqrt{2}} = 20 ]Let me denote ( frac{k q^2}{s} = U_s ), so:[ 4 U_s + 2 times frac{U_s}{sqrt{2}} = 20 ]Which simplifies to:[ 4 U_s + sqrt{2} U_s = 20 ][ U_s (4 + sqrt{2}) = 20 ][ U_s = frac{20}{4 + sqrt{2}} ]As before.So, the total energy in the new configuration is:[ left(frac{7 sqrt{2}}{2} + 2right) U_s ]Substitute ( U_s = frac{20}{4 + sqrt{2}} ):[ left(frac{7 sqrt{2}}{2} + 2right) times frac{20}{4 + sqrt{2}} ]Let me compute this:First, simplify ( frac{7 sqrt{2}}{2} + 2 ):[ frac{7 sqrt{2} + 4}{2} ]So, the expression becomes:[ frac{7 sqrt{2} + 4}{2} times frac{20}{4 + sqrt{2}} ]Multiply numerator and denominator:[ frac{(7 sqrt{2} + 4) times 20}{2 times (4 + sqrt{2})} ]Simplify:[ frac{20(7 sqrt{2} + 4)}{2(4 + sqrt{2})} = frac{10(7 sqrt{2} + 4)}{4 + sqrt{2}} ]Now, let's rationalize the denominator by multiplying numerator and denominator by ( 4 - sqrt{2} ):[ frac{10(7 sqrt{2} + 4)(4 - sqrt{2})}{(4 + sqrt{2})(4 - sqrt{2})} ]Compute the denominator:[ (4)^2 - (sqrt{2})^2 = 16 - 2 = 14 ]Now, expand the numerator:[ (7 sqrt{2} + 4)(4 - sqrt{2}) = 7 sqrt{2} times 4 + 7 sqrt{2} times (-sqrt{2}) + 4 times 4 + 4 times (-sqrt{2}) ][ = 28 sqrt{2} - 7 times 2 + 16 - 4 sqrt{2} ][ = 28 sqrt{2} - 14 + 16 - 4 sqrt{2} ]Combine like terms:[ (28 sqrt{2} - 4 sqrt{2}) + (-14 + 16) ][ = 24 sqrt{2} + 2 ]So, the numerator becomes:[ 10(24 sqrt{2} + 2) = 240 sqrt{2} + 20 ]Therefore, the total energy is:[ frac{240 sqrt{2} + 20}{14} = frac{240 sqrt{2}}{14} + frac{20}{14} ]Simplify:[ frac{120 sqrt{2}}{7} + frac{10}{7} = frac{120 sqrt{2} + 10}{7} ]Which matches what I had earlier.So, the total energy in the new configuration is ( frac{120 sqrt{2} + 10}{7} ) Joules.Now, to find the increase in energy, subtract the initial energy of 20 Joules:[ Delta U = frac{120 sqrt{2} + 10}{7} - 20 ]Convert 20 to a fraction with denominator 7:[ 20 = frac{140}{7} ]So:[ Delta U = frac{120 sqrt{2} + 10 - 140}{7} = frac{120 sqrt{2} - 130}{7} ]This is the exact expression for the increase in energy.But let's see if we can simplify this further or express it in a more elegant form.Alternatively, we can factor out a common term:[ Delta U = frac{10(12 sqrt{2} - 13)}{7} ]But I'm not sure if this is any better.Alternatively, we can compute the numerical value to get an approximate idea.Compute ( 120 sqrt{2} approx 120 times 1.4142 approx 169.704 )So:[ Delta U approx frac{169.704 - 130}{7} = frac{39.704}{7} approx 5.672 text{ Joules} ]So, approximately 5.67 Joules more energy is stored.But the problem asks for the exact value, so I should present it in terms of ( sqrt{2} ).Therefore, the increase in energy is:[ frac{120 sqrt{2} - 130}{7} text{ Joules} ]Alternatively, factor out a 10:[ frac{10(12 sqrt{2} - 13)}{7} text{ Joules} ]But I think the first form is acceptable.So, the final answer is:[ boxed{frac{120 sqrt{2} - 130}{7} text{ Joules}} ]